Plane Trees Research Articles

The height of multiple edge plane trees

Multi-edge trees as introduced in a recent paper of Dziemia\'nczuk are plane trees where multiple edges are allowed. We first show that $d$-ary multi-edge trees where the out-degrees are bounded by $d$ are in bijection with classical $d$-ary trees. This allows us to analyse parameters such as the height. The main part of this paper is concerned with multi-edge trees counted by their number of edges. The distribution of the number of vertices as well as the height are analysed asymptotically.

Hopf algebras and topological recursion

We consider a model for topological recursion based on the Hopf algebra of planar binary trees defined by Loday and Ronco (1998 Adv. Math. 139 293–309 We show that extending this Hopf algebra by identifying pairs of nearest neighbor leaves, and thus producing graphs with loops, we obtain the full recursion formula discovered by Eynard and Orantin (2007 Commun. Number Theory Phys. 1 347–452).

Analysis of the physiological responses of the plane and willow trees against air pollution in Tehran .

Trees can improve air quality, but at the same time, pollution might put them at risk. Plane(Platanus orientalis L) and willow trees (Salix babylonicaL) are important trees in the green space of Tehran. Plane trees in Tehran have experienced early autumn in recent years due to severe air pollution. The present study, investigated some physiological traits of these trees under study to determine the cause of early autumn phenomenon. Three regions in Tehran were chosen for the study: Sadra Park as a clean region, Al-Mahdi Park as the polluted region 1, and Avesta Park as the polluted region 2. The results showed that the highest levels of chlorophyll a, chlorophyll b, total chlorophyll, carotenoids, and anthocyanins were observed in the leaves of the plane trees of Sadra Park and the leaves of the willow trees of the Avesta Park and Sadra Park. The average concentration of lead (Pb) in the plane leaves was observed in the Avesta Park and the highest cadmium concentration was recorded for the Sadra Park while the concentration of these two metals in the leaves of willow trees in the three regions were not significantly different. This may suggest that the willow trees, by an increase in the amount of anthocyanins of their leaves, prevent the reduction of the amount of photosynthetic pigments under the influence of air pollution. By late spring, the two plants could relatively preserve similar amounts of lead and cadmium in their leaves.

The Plane Trees of Diomedes: Staging the Islands of the Blessed in the Adriatic Sea

Taking its starting point from the current theoretical debate about “religious aesthetics” (Religionsästhetik), this article offers a new interpretation of the landscaping of the Island of Diomedes. On the basis of a survey of the primary sources, it demonstrates that the descriptions of the Island of Diomedes have to be seen in the context of the mythological motif of the Islands of the Blessed. This in turn suggests an explanation for Theophrastus’ statement that in the Adriatic region of his day, the plane tree grew only around the shrine of Diomedes: given the typical associations of the plane tree, the planting of planes on the sanctuary island of Diomedes may have been intended to bring the appearance of the Island of Diomedes in line with contemporary stereotypes about the appearance of the Islands of the Blessed; thus, the plane trees of Diomedes constitute a historical case of direct relevance for the question of “religious aesthetics.” The article goes on to demonstrate that the plane trees of Diomedes constitute conclusive evidence against the identification of the Island of Diomedes with the island of Pelagosa (recently suggested on the basis of archaeological finds). Instead, the Island of Diomedes should be sought among the Isole Tremiti.

On the complexity of role colouring planar graphs, trees and cographs

We prove several results about the complexity of the role colouring problem. A role colouring of a graph G is an assignment of colours to the vertices of G such that two vertices of the same colour have identical sets of colours in their neighbourhoods. We show that the problem of finding a role colouring with 1<k<n colours is NP-hard for planar graphs. We show that restricting the problem to trees yields a polynomially solvable case, as long as k is either constant or has a constant difference with n, the number of vertices in the tree. Finally, we prove that cographs are always k-role-colourable for 1<k≤n and construct such a colouring in polynomial time.

One-point functions in defect CFT and integrability

We calculate planar tree level one-point functions of non-protected operators in the defect conformal field theory dual to the D3-D5 brane system with k units of the world volume flux. Working in the operator basis of Bethe eigenstates of the Heisenberg XXX_{1/2} spin chain we express the one-point functions as overlaps of these eigenstates with a matrix product state. For k=2 we obtain a closed expression of determinant form for any number of excitations, and in the case of half-filling we find a relation with the N\'eel state. In addition, we present a number of results for the limiting case of infinite k.

Rate enhancement via sodium dodecyl sulfate (SDS) encapsulation of metal-mediated cerium(IV) oxidation of d-mannitol to d-mannose at room temperature and pressure: a kinetic and mechanistic approach

d-mannitol is a sweet-tasting hexitol widely distributed in nature, and is found in olive trees, plane trees, fruits, and vegetables. However, production of mannitol and mannose by extraction of plant raw materials is no longer economical. Selective oxidation of d-mannitol gives d-mannose. Oxidation kinetics of d-mannitol by Ce(IV) in an aqueous medium in the presence of sodium dodecyl sulfate (SDS) has been carried out to observe the micellar effect on rate. The oxidation kinetics was studied by UV–Vis spectrophotometry. Five different metal ions Cr(III), Mn(II), Fe(II), Cu(II), and Ag(I) are used. Both Cr(III) and Mn(II) are active catalysts for the d-mannitol oxidation in the presence and absence of anionic surfactants. The substrate undergoes effective collision with the expected positive reactive species Ce(SO4)2+ resulting enhancement of rate. The presence of SDS surfactant was found to accelerate the reaction rate and this effect has been explained by the partitioning of the reactants in micelle. The effect of sodium dodecyl sulfate (SDS) on rate also indicate that Ce(SO4)2+ was the main reactive form of cerium(IV). The main product d-mannose was identified by the FTIR spectroscopy and spot test. The combination of Mn(II) and SDS is the most suitable combination for the conversion of d-mannitol to d -mannose.

Chebyshev Polynomials, Zolotarev Polynomials, and Plane Trees

A polynomial with exactly two critical values is called a generalized Chebyshev polynomial (or Shabat polynomial). A polynomial with exactly three critical values is called a Zolotarev polynomial. Two Chebyshev polynomials f and g are called Z-homotopic if there exists a family pα, α\( \epsilon \) [0, 1], where p0 = f, p1 = g, and pα is a Zolotarev polynomial if α\( \epsilon \) (0, 1). As each Chebyshev polynomial defines a plane tree (and vice versa), Z-homotopy can be defined for plane trees. In this work, we prove some necessary geometric conditions for the existence of Z-homotopy of plane trees, describe Z-homotopy for trees with five and six edges, and study one interesting example in the class of trees with seven edges.

Enumeration of Weighted Plane Trees

In weighted trees, all edges are endowed with positive integral weight. We enumerate weighted bicolored plane trees according to their weight and number of edges.

Weighted Trees with Primitive Edge Rotation Groups

Let R, S ∈ ℂ[x] be two coprime polynomials of the same degree with prescribed multiplicities of their roots. A classical problem of number theory actively studied during the last half-century is, What could be the minimum degree of the difference T = R − S? The theory of dessins d’enfants implies that such a minimum is attained if and only if the rational function f = R/T is a Belyi function for a bicolored plane map all of whose faces except the outer one are of degree 1. Such maps are called weighted trees, since they can be conveniently represented by plane trees whose edges are endowed with positive integral weights.

Enumeration of One Class of Plane Weighted Trees

We present an enumeration formula for weighted trees, i.e., trees where vertices and edges have weights (a weight is a positive integer) and the weight of each vertex is equal to the sum of the weights of the edges incident to it. Each tree has a binary structure: we can color its vertices in two colors, black and white, so that adjacent vertices have different colors. In this work, the following problem is considered: enumerate weighted plane trees with given sets of weights of black and white vertices.

MICROPROPAGATION OF MATURE PLATANUS × HISPANICA TREES BY AXILLARY SHOOT PROLIFERATION

A protocol for vegetative propagation by organogenesis has been developed for several anthracnose tolerant Platanus A— hispanica mature trees. Different types of putative juvenile and mature plant materials were used. From them apical and nodal explants were tested to initiate cultures. Nodal explants from well-developed epicormic shoots, flushed from pieces of branches, gave low contamination percentages and 42% of established explants. Newly formed shoots were multiplied in MS medium supplemented with 1.33 µM BAP and 0.27 µM NAA. The effect of using both gelled and liquid medium was tested. Multiplication rate was highly dependent on genotype. The highest multiplication rate was obtained on gelled medium (4 new shoots per explants each 8 weeks). Microshoots were rooted on a MS gelled medium with macronutrients at half, reduced sucrose and supplemented with 0.49 µM IBA. Rooting ranged between 30 and 80% depending on genotype. Two major problems in the micropropagation of Spanish plane trees were observed: leaf and shoot tip necrosis, and low plantlet survival during acclimatisation. Overall, losses of explants in proliferation due to necrosis were more pronounced in liquid medium (69%) than on solid medium (57%). Only a very few plants could be acclimatised.

Economic Valuation of Urban Trees: Ribnjak Park Case Study, Zagreb

Background and Purpose: Population growth, urbanisation and technological development are creating a growing need for urban forests and parks, which are becoming green oases for recreation and relaxation. Apart from the sociological and economic components, urban forest valuation is presented through tourism, the market value of main and secondary forest products, and the growing value of real estate in the vicinity of green areas. Environmental economics explores the optimal ratio between the costs and the benefits received from the investment in the environment. The aim of this research is monetary valuation of urban trees. Materials and Methods: A Danish model for tree value determination was applied in Ribnjak Park as a case study. The model is based on tree growing costs and the present value. It is limited by the subjective aesthetic tree value estimation, but it is used in Europe because of its practicality. Individual tree value estimation is used because of the tree damage from vehicles or new residential buildings. The method is suitable for individual trees or groups of trees, but it is not appropriate for forest stands. Twenty random selected trees from nine different tree species have been analysed in the park. Diameter at breast height, tree height, expected age, aesthetic value and location were recorded for each tree. Furthermore, ecological, social and health tree values were taken into account separately with the calculation of points. Results: According to the evaluation, the average monetary value of one tree in Ribnjak Park is 542 EUR. The average diameter at breast height is 57.86 cm with the average age of 96.14 years. Plane trees have the highest value in comparison to other sampled species. Conclusions: Tree values vary depending on age, dimension or aesthetic values. The disadvantage of this method is in the estimation of very old tree value and in high involvement of personal estimation, which creates an opportunity for future development of the model and for its modification. The calculation of urban forest values is a complex process because of its impossibility to do a market evaluation of all of the benefits that could be considered as public wealth.

Navigation in tree spaces

The orientable cover of the moduli space of real genus zero algebraic curves with marked points is a compact aspherical manifold tiled by associahedra, which resolves the singularities of the space of phylogenetic trees. The resolution maps planar metric trees to their underlying abstract representatives, collapsing and folding an explicit geometric decomposition of the moduli space into cubes, endowing the resolving space with an interesting canonical pseudometric. Indeed, the given map can be reinterpreted as relating the real and the tropical versions of the Deligne–Knudsen–Mumford compactification of the moduli space of Riemann spheres.

Pests and pathogen threats to plane (Platanus) in Britain

Plane trees (Platanus spp.) are a valuable part of the urban landscape in Britain, providing many ecosystem services. This commonly planted street tree tends to be very tolerant of the harsh conditions imposed by urban environments but there is increasing awareness of a number of pests and pathogens that affect Platanus. Some such as Apiognomonia veneta, the causal agent of anthracnose of plane, and decay fungi such as Inonotus hispidus and Perenniporia fraxinea are native to Britain, but the origin of Splanchnonema platani is not yet clear and other organisms such as Ceratocystis platani, the cause of canker stain, and decay fungus Inonotus rickii are not present in this country. As the impact of some of these organisms on both trees and the overall landscape can be severe, the aim of this review was to increase awareness of their appearance, potential impact and any available management or control options.

Bootstrapping and double-exponential limit laws

Combinatorics We provide a rather general asymptotic scheme for combinatorial parameters that asymptotically follow a discrete double-exponential distribution. It is based on analysing generating functions Gh(z) whose dominant singularities converge to a certain value at an exponential rate. This behaviour is typically found by means of a bootstrapping approach. Our scheme is illustrated by a number of classical and new examples, such as the longest run in words or compositions, patterns in Dyck and Motzkin paths, or the maximum degree in planted plane trees.

Sycamore Tree Lace Bug (Corythucha ciliataSay) (Hemiptera: Tingidae) Reaches Africa

In February 2014 colonies of a large and conspicuous lacebug (Fig. 1A) were collected from the undersides of the leaves of London plane trees (Platanus x acerifolia) in the suburb of Newlands, Cape Town, South Africa. Infected leaves could easily be identified by their characteristic bronzed appearance (Fig. 1B). The lace bugs were subsequently identified as Corythucha ciliata (Say) using keys to North American Corythucha species (Mead 1989) and economically important tingids of the world (Stonedahl et al. 1992). Corythucha species have high host-specificity, whilst C. ciliata is the only lace bug known to feed on Platanus species (sycamores) - this association is thus considered to be diagnostic for the species (CABI 2014). C. ciliata has a broad native range across the eastern parts of North America and Canada, where it is largely monophagous on Platanus (sycamore) trees, with P. occidentalis being its main host. However, it also occurs on Platanus hybrids, and in invasive parts of its range feeds on P. orientalis and P. x acerifolia (the hybrid between P. occidentalis and P. orientalis (CABI 1984).

The Trees of the Sacred Natural Sites of Zagori, NW Greece

Cultural landscapes can often be identified by the presence of sacred trees which have been retained and can be recognised as distinctive veteran trees. The characteristics of these trees such as huge size and longevity give them an enduring presence in the landscape, while their conceptualisation as the ‘domesticated’ wild, transform them into symbols of spirituality and local history. In Zagori, NW Greece, trees gain sanctity by virtue of proximity or connection with sacred sites often associated with churches. In these sites, trees can grow into natural shapes as a result of strong taboos, which prevent use for private needs. There is an association between tree species and the nature of the sacred site: broadleaved oaks and maples are associated with outlying churches; plane trees are located in central squares, next to the church and provide a focal point for community life, while in cemeteries native evergreens are nowadays replaced by planted conifers. In the present day local communities appreciate sacred trees as living elements of their collective memory and local history.

Brown's Theorem and its Application for Enumeration of Dissections and Planar Trees

We present new functional equations connecting the counting series of plane and planar (in the sense of Harary and Palmer) dissections. Simple rigorous expressions for counting symmetric $r$-dissections of polygons and planar $S$-dissections are obtained.

在平面二元树族上的量子电动力学的H-Hopf模

利用费曼图和平面二元树的结合，定义了QED的重正化群并指出群上半直积的H-Hopf模和H-Hopf模的半直余积，平面二元树上的H-Hopf模的传播及带电的H-Hopf模，最后得出量子电动力学的H-Hopf模及电子和量子的重正化余作用。 Using the combination of Feynman diagrams and the planar binary tree, the renormalization group of QED was defined and the H-Hopf module of semidirect product of the group, the semidirect coproduct of the H-Hopf module, the spread of the H-Hopf module on the planar binary tree and the H-Hopf module with charge were pointed out. Finally it concluded the H-Hopf module of quantum electrodynamics and the renormalization coaction of electron and photon.